The U.S. routinely has more troops on the ground in Iraq than the 3,500-3,600 frequently cited by Defense Secretary Ashton Carter, the high command and President Barack Obama, a U.S. military spokesman in Baghdad said Wednesday.

"It's fair to say" that the number of U.S. troops in Iraq serving as trainers and advisors -- or in support or on special assignment -- was well above 4,000 on a daily basis, said Army Col. Steve Warren, a spokesman for Combined Joint Task Force-Operation Inherent Resolve led by Army Lt. Gen. Sean MacFarland.

In a video briefing from Baghdad to the Pentagon, Warren essentially confirmed a Daily Beast report that the current number of U.S. troops in Iraq was about 4,450. "I don't think we're going to dispute what's in the Daily Beast article," he said.

Warren said U.S. troops on the ground also exceeded the "cap" of about 3,870 agreed to with the Baghdad government of Prime Minister Haider al-Abadi. "We're aware of those restraints and limitations," he said.

Warren attributed some of the excess to "overlap" in troop rotations, noting that the 101st Airborne Division currently was in the process of switching out with units of the 82nd Airborne Division. "So there's naturally going to be overlap there. There's this continuous churn and turnover" of troops coming and going, he said.

"Additionally there's always people cycling through here" on temporary duty, or TDY, status, Warren said. He cited a trip to Iraq of the Defense Department's Inspector General, who had a security detail of about 25 American troops.

The "boots on the ground" issue in the effort in Iraq and Syria to defeat the Islamic State has been a source of friction with Congress. The issue has also been a main topic of the presidential campaigns since Obama authorized the deployment of small units to protect the U.S. Embassy in Baghdad when fighters affiliated with the Islamic State in Iraq and Syria, or ISIS, swept into Iraq in the late spring of 2014. U.S. airstrikes against ISIS began in the country in early August 2014.

Obama has consistently ruled out ground combat for U.S. troops, limiting their roles to training, advising and assisting, but critics of the administration have warned of "mission creep" in the growing number of Americans on the ground.

In recent weeks, Carter has said repeatedly that he was looking for "additional opportunities" for the deployment of more U.S. troops as trainers and advisors in an effort to retake Mosul, the second largest city in Iraq, and also to oust ISIS from Raqqa, the self-proclaimed insurgent capital in northeastern Syria.

Carter has also said that decisions on additional deployments would likely come after a meeting with coalition and NATO allies in Brussels later this month to determine the willingness of the allies to make their own contributions.

In a previous briefing to the Pentagon from Baghdad, Warren said that the number of additional U.S. troops would likely number in the "hundreds, not thousands."

In Wednesday's briefing, Warren said that more deployments were also conditional on the agreement of the Iraqi government. U.S. advice on the number of U.S. troops needed in Iraq does not come "without that political reality," he said. "We understand that there's going to be a political factor here," but "there's not a big disconnect now. It's their [the Iraqis'] operations that we're supporting."